Integracija metodom Monte Carlo

master's thesis

• Title (english): Monte Carlo integration
• Author: Matea Debak
• Mentor: Snježana Lubura Strunjak (mentor)
• Committee member: Snježana Lubura Strunjak (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pri rješavanju problema u brojim područjima znanosti nerijetko se pojavljuju integrali čije je vrijednosti potrebno izračunati. Međutim, sam proces integracije složen je te ponekad vrijednost integrala nije moguće odrediti egzaktno. Iako postoje različite numeričke metode integracije koje pružaje efikasne algoritme za aproksimativno rješenje, u određenim slučajevima, primjerice u višedimenzionalnim problemima, zbog nedovoljne brzine ili nepraktičnosti izvođenja one se mogu bitno zakomplicirati. U ovom radu opisali smo jednu drugačiju metodu integracije - Monte Carlo metodu, kojom u pravilu možemo dobiti dobre procjene vrijednosti integrala i ocijeniti njihovu grešku. Ova aproksimativna metoda koristi se u slučajevima kada problem nije moguće (ili nije jednostavno) riješiti analitički, a posebno je korisna u rješavanju višedimenzionalnih problema jer točnost procjene ne ovisi o dimenziji prostora. Stoga je primjena ove metode u novije vrijeme sasvim uobičajena i poželjna. Primarni cilj ovog rada bio je opisati ideju integracije metodom Monte Carlo i teoretski ju razraditi. Zaključili smo kako se cijeli proces integracije ovom metodom temelji na mogućnosti efikasnog generiranja slučajnih varijabli iz određene distribucije te naveli neke od metoda generiranja. Također, pokazali smo kako je u tom procesu integracije ipak potrebno pažljivo odabrati Monte Carlo procjenitelj, odnosno distribuciju iz koje ćemo generirati slučajne varijable. Naime, upravo o tom izboru ovisi varijanca procjenitelja koju želimo maksimalno smanjiti. Taj proces smanjivanja varijance Monte Carlo procjenitelja opisali smo metodom uzorkovanja po važnosti (Importance Sampling) i zaključili da ta metoda daje bolju procjenu vrijednosti integrala od ”naivne” Monte Carlo metode.

Abstract (english)

Solving problems in multiple areas of science often requires calculating the values of integrals. However, the process of integration is so complex that sometimes it is not possible to determine the exact value of an integral. There are various numerical integration methods that offer efficient algorithms for calculating the approximate solution. Still, in certain cases (e.g. multidimensional problems), due to insufficient speed or impracticality of the performance they can get significantly complicated. In this thesis, we described a different method of integration - the Monte Carlo method. This method allows us to get appropriate estimated values of integrals and their approximation error. This approximate method is also used in situations when the problem is too difficult (or impossible) to solve analytically. It is especially useful when solving multidimensional problems because the accuracy of the estimated value does not depend on the dimension of space. Hence, using this method is both common and desirable in modern times. Primary goal of this thesis was to describe Monte Carlo integration method and to elaborate on it theoretically. It was concluded that the whole process of using this method is based on the idea of efficiently generating random variables from a certain distribution. In addition, we named some of the methods used for generating such variables. It is worth pointing out that it is necessary to carefully select the Monte Carlo estimator i.e. the distribution from which we generate random variables. Namely, it is on that selection of the distribution that the variance of the estimator which we are trying to maximally reduce depends on. This process of reduction of the variance of the Monte Carlo estimator was described using the Importance Sampling method which has proven to give a better estimated value of the integral than the “naive” Monte Carlo method.

• Keywords: procjena vrijednosti integrala; generiranje slučajnih varijabli; višedimenzionalni problemi; metoda uzorkovanja po važnosti
• Keywords (english): values of integrals; generating random variables; multidimensional problems; Importance Sampling method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:635219
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-10-26 10:48:09